The Bundesliga side raced into an early lead through Reus and Henrikh Mkhitaryan, before Robert Lewandowski twice responded to second-half goals from Russians to put paid to their comeback efforts.
Reus was proud of the way BVB performed in the clash in Russia, but warned that the goals from Oleg Shatov and Hulk after the break proved that Jurgen Klopp's men cannot afford to sit back in the return leg at Signal Iduna Park.
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"We started very well. Our tactics worked perfectly, we took an early 2-0 lead and put in a fine display, especially in the first half," the Germany international told Uefa's official website.
"In the second period we slacked off a bit and this should serve as a lesson to us for the return leg. Our 2-0 advantage was deserved, as was the final result, I think.
"I always say it doesn't matter what the individual does, but to win as a team, and that's what we have shown today.
"We should take it game by game. Today we took the first step towards the next round, but you could also see that an away team is always capable of scoring.
"We did a lot of the work today and so a lot of things would have to happen for us not to go through in the return leg. Nevertheless, we have to be on our guard.
"I am, of course, feeling good about my performance. It was important that the team won and we are very happy.
"We will take it one match at a time. In a couple of weeks we have the return leg in our stadium and we want to go through then."
Defender Lukasz Piszczek was pleased with his side's reaction following the humbling 3-0 Bundesliga defeat to Hamburg at the weekend, but reserved special praise for Lewandowski, who will join Bayern Munich in the summer.
"We had quite a good start with the two goals and that made life a bit easier for us. But we were also very focused the whole time and were a bit unlucky to concede those two goals," he said.
"I don't think we were lacking concentration when we conceded, those two goals were a bit of a fluke for Zenit. We stood up well and were unlucky in those situations. We did well and I hope we play as well in the return leg.
"Robert [Lewandowski] is a world-class player. We are happy he is still with us and hope he will continue to play as well as he did today and over the past few weeks.
"We had to show a reaction after that match in Hamburg, as there were a lot of mistakes. It was not good for us, but we knew we had a very important match coming up and strongly focused on getting a very good result. We managed that."